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Honors Calculus Summer Work and List of Topical Understandings In order to be a successful student in the Honors Calculus course, it is imperative to practice and improve your current math skills. Calculus is a course in advanced algebra and geometry that is used to study how things change. To be successful in this course, students must be highly motivated, dedicated, capable learners who put forth great effort in working independently to fully understand the concepts in the honors calculus curriculum. The summer packet will act as a refresher of math concepts that should already be mastered when you enter the honors calculus course in September. This packet is a requirement for students entering honors calculus and is due on the first day of class. Requirements for completing summer packet: You must complete each of the problems in the packet. You must show all of your work in the space provided on the packet. Be sure all problems are neatly organized and all writing is legible. In the event that you are unsure how to perform functions on your calculator, you may need to read through your calculator manual to understand the necessary syntax or keystrokes. You must be familiar with certain built-in calculator functions such as finding values, intersection points, using tables, and zeros of a function. I expect you to come in with certain understandings that are prerequisite to Calculus. A list of these topical understandings is below. Summer Work Topics: Factoring Zeros/roots/x-intercepts of rational and polynomial functions Polynomial Long Division Completing the square Write the equation of a line Quadratic formula Unit Circle Composite function and notation Solving trigonometric equations Domain/Range Interpreting and comprehending word problems Trigonometric Identities (listed at the end of the summer packet) Graphing, simplifying expressions, and solving equations of the following types: Trigonometric Exponential Rational Polynomial Piecewise Power Logarithmic Radical Finally, I suggest not waiting until the last two weeks of summer to begin on this packet. If you spread it out, you will most likely retain the information much better. Once again this is due, completed with quality, on the first day of class. It is your ticket into the class. An extra copy of the work can be found on the High School Web Site under Student Resources. Name: _________________________ Honors Calculus Summer Math Packet 7 x9 5 x6 1) Simplify . Express your answer using a single radical. 2). Factor completely. 3) Determine the domain, range and the zeros of: f ( x) 13 20 x x 2 3x 4 . 6 x3 17 x 2 5 x Domain: __________________ Range: __________________ Zeros: __________________ 4) Find the equation of the line through ( 2, 7) and (3,5) in point slope form. 5) Rewrite the expression log5 ( x 3) into an equivalent expression using only natural logarithms 6) Solve the equation both algebraically and graphically. 4x 3 5 x 4 Algebraic Solution: Graph Solution: 8) Three sides of a fence and an existing wall form a rectangular enclosure. The total length of a fence used for the three sides is 240 ft. Let x be the length of two sides perpendicular to the wall as shown. Write an equation of area A of the enclosure as a function of the length x of the rectangular area as shown in the above figure. The find value(s) of x for which the area is 5500 ft 2 ? x x Existing wall 9) Let f ( x) x 3 and g ( x) x2 1 . Compute ( g f )( x) , state its domain in interval notation. ( g f )( x) : _______________________ Domain: 10) Let f ( x) 3x 7 . Find f 1 ( x) , the inverse of f ( x) x2 _______________________ 11) Find an equation for the parabola whose vertex is (2, 5) and passes through (4, 7) . Express your answer in the standard form for a quadratic. 12) Which of the following could represent a complete graph of f ( x) ax x3 , where a is a real number? A. B. C. D. 13) Find a degree 3 polynomial with zeros -2, 1, and 5 and going through the point (0, 3) . 14) The graph of y 2 a x 3 for a 1 is best represented by which graph? A. B. C. D. 15) Describe the transformations that can be used to transform the graph of f ( x) ln x to a graph of f ( x) 4 ln( x 2) 3 . Graph each function in the designated area below. f ( x) ln x f ( x) 4 ln( x 2) 3 16) 17) 18) 19) The number of elk after t years in a state park is modeled by the function P(t ) a) What was the initial population of elk? b) When will the number of elk be 750? c) Use your calculator to determine the maximum number of elk possible in the park? Arturo invests $2700 in a savings account that pays 9% interest, compounded quarterly. If there are no other transactions, when will his balance reach $4550? Simplify csc( x) tan( x) sin( x)cos( x) A. sin( x) cos 2 ( x) B. cos( x) sin 2 ( x) C. sin 2 ( x) cos( x) D. cos2 ( x) sin( x) Find the exact value of each without the use of a calculator. sin 3 20) 1216 . 1 75e 0.03t 3 cos 2 5 tan 6 2 csc 3 cot 2 æ æ p öö Without using a calculator, find the exact value of cos-1 ç cos ç ÷÷ . Justify your answer. è è 6 øø 21) Solve the inequality: x 2 x 12 0 . A. (, 4) (3, ) B. x 4, x 3 C. (3, 4) D. (, 3) (4, ) 22) Find the perimeter of a 30 slice of cheesecake if the radius of the cheesecake is 8 inches. 23) x 3 7 x 2 14 x 8 Use polynomial long division to rewrite the expression x4 24) Transform y = -3x2 – 24x + 11 to vertex form. 25) Two students are 180 feet apart on opposite sides of a telephone pole. The angles of elevation from the students to the top of the pole are 35 and 23 . Find the height of the telephone pole. 26) Graph the piecewise function. x 2 2 x 1 f ( x) 2 x 1 3 x 5 1 x 3 27) Solve the equation 2sin 2 ( x) cos( x) cos( x) algebraically. 28) Find all the exact solutions to 2sin 2 ( x) 3sin( x) 2 0 on the interval 0, 2 . 29) For the function f ( x) graphed answer the following 30) A. f (3) B. f ( x) 0 C. f ( 0) D. f ( x) 1 Given that f ( x) x5 . Graph the function. Find the asymptotes and the domain of the function. x2 Domain: __________ Vertical Asymptote(s): _________ Horizontal Asymptote(s): ________ 31) Use a graphing calculator to approximate all of the function’s real zeros. Round your results to four decimal places. f ( x) 3x6 5x5 4 x3 x 2 x 1 32) Factor to solve the inequality. Write your answer in interval notation. 0 33) Simplify the expression as much as possible 35) Simplify the rational expression. x3 64 x4 1 1 3( x h) 3x h 2(2 x 1) 2 x 2 x x6 x3 2 x 36) Use the quadratic formula to find the exact solution to x4 – 5x2 + 3 = 0. Show all work. Graph the following problems. You should know the basic shape of each graph and how to shift each graph. 37) y=x 38) y = x2 40) y = x3 41) 44) 43) y = │x – 3 │ 39) y = 2(x – 1)2 – 3 y = 3x 42) y= y = log x 45) y = ex x2 TRIG IDENTITY REFERENCE SHEET Reciprocal Properties tan x 1 , cot x cot x sin x csc x 1 1 , tan x sin x cos x sec x 1 1 , csc x csc x 1 , sin x tan x cot x 1 Quotient Properties tan x sin x , cos x cot x cos x sin x Pythagorean Properties cos 2 x sin 2 x 1 1 tan 2 x sec 2 x cot 2 x 1 csc2 x Odd and Even Function Properties sin x sin x csc x csc x cos x cos x sec x sec x tan x tan x cot x cot x Co-function Properties sin x cos x 2 cos x sin x 2 csc x sec x 2 sec x csc x 2 tan x cot x 2 cot x tan x 2 cos x 1 , sec x sec x 1 cos x Sum and Difference Properties (Composite Argument Properties) 1. sin u v sin u cos v cos u sin v 4. cos u v cos u cos v sin u sin v 2. sin u v sin u cos v cos u sin v 5. cos u v cos u cos v sin u sin v 3. tan u v tan u tan v 1 tan u tan v 6. tan u v tan u tan v 1 tan u tan v Double Angle Formulas 1. sin 2 A 2sin A cos A 2. cos 2 A cos 2 A sin 2 A or using Pythagorean substitutions, cos 2 A 1 2sin 2 A cos 2 A 2 cos 2 A 1 3. tan 2 A 2 tan A 1 tan 2 A Half-Angle Formulas sin A 1 cos A 2 2 tan A 1 cos A , 2 1 cos A tan A sin A , 2 1 cos A cos A 1 cos A 2 2 tan A 1 cos A 2 sin A